Idea

It is a hypercover satisfying an extra condition that roughly says that it is degreewise freely given by representables.

Definition

Regard X∈CX \in C under the Yoneda embedding as an object X∈[Cop,sSet]proj,locX \in [C^{op}, sSet]_{proj,loc}. Then a morphism (Y→X)∈[Cop,sSet](Y \to X) \in [C^{op}, sSet] is a split hypercover of XX if

YY is split in that the image of the degeneracy maps identifies with a direct summand in each degree.

Properties

The splitness condition on the hypercover is precisely such that YY becomes a cofibrant object in [Cop,sSet]proj,loc[C^{op}, sSet]_{proj,loc}, according to the characterization of such cofibrant objects described here.